152
6. Quantum Field Theory II: Interacting Scalar Fields
6.2 Perturbation theory for interacting fields: the Dyson
expansion of the S-matrix
On the third day of the journey a remarkable thing happened; going into
a sort of semi-stupor as one does after 48 hours of bus-riding, I began to
think very hard about physics, and particularly about the rival radiation
theories of Schwinger and Feynman. Gradually my thoughts grew more
coherent, and before I knew where I was, I had solved the problem that
had been in the back of my mind all this year, which was to prove the
equivalence of the two theories.
—From a letter from F. J. Dyson to his parents, 18 September 1948, as
quoted in Schweber (1994), p 505.
For definiteness, let us consider the Lagrangian
ˆ 1 ˆ
2 ˆ
φ
3
L = ∂ μ φ∂
μ φ ˆ −
1 m φ
2
− λφ ˆ 3 ≡ L ˆ KG − λ ˆ
(6.12)
2
2
1
1 2 ˆ
with λ > 0. Equation (6.12) is like ‘L ˆ = T ˆ −V ˆ ’ where V ˆ = (∇φ ˆ )
2 + m φ
2 +
2
2
λφ ˆ3 is the ‘potential’. Though simple, this Lagrangian is unfortunately not
physically sensible. The classical particle analogue potential would have the
form V (q) =
1 ωq
2 + λq
3 . If we sketch V (q) as a function of q we see that,
2
for small λ, it retains the shape of an oscillator well near q = 0, but for q
sufficiently large and negative it will ‘turn over’, tending ultimately to −∞ as
q → −∞. Classically we expect to be able to set up a successful perturbation
theory for oscillations about the equilibrium position q = 0, provided that
the amplitude of the oscillations is not so large as to carry the particle over
the ‘lip’ of the potential; in the latter case, the particle will escape to q =
−∞, invalidating a perturbative approach. In the quantum mechanical case
the same potential V (q) is more problematical, since the particle can tunnel
through the barrier separating it from the region where V → −∞. This
means that the ground state will not be stable. An analogous disease affects
the quantum field case – the supposed vacuum state will be unstable, and
indeed the energy will not be positive-definite.
Nevertheless, as the reader may already have surmised, and we shall confirm later in this chapter, the ‘φ-cubed’ interaction is precisely of the form
relevant to Yukawa’s exchange mechanism. As we have seen in the previous section, such an interaction will typically give rise to matrix elements
between one-quantum and two-quantum states, for example, exactly like the
basic Yukawa emission and absorption process. In fact, all that is necessary to make the φ ˆ3 -type interaction physical is to let it describe, not the
‘self-coupling’ of a single field, but the ‘interactive coupling’ of at least two
different fields. For example, we may have two scalar fields with quanta ‘A’
φ
2 ˆ
and ‘B’, and an interaction between them of the form λ ˆ φ B . This will allow
A
6. Quantum Field Theory II: Interacting Scalar Fields
6.2 Perturbation theory for interacting fields: the Dyson
expansion of the S-matrix
On the third day of the journey a remarkable thing happened; going into
a sort of semi-stupor as one does after 48 hours of bus-riding, I began to
think very hard about physics, and particularly about the rival radiation
theories of Schwinger and Feynman. Gradually my thoughts grew more
coherent, and before I knew where I was, I had solved the problem that
had been in the back of my mind all this year, which was to prove the
equivalence of the two theories.
—From a letter from F. J. Dyson to his parents, 18 September 1948, as
quoted in Schweber (1994), p 505.
For definiteness, let us consider the Lagrangian
ˆ 1 ˆ
2 ˆ
φ
3
L = ∂ μ φ∂
μ φ ˆ −
1 m φ
2
− λφ ˆ 3 ≡ L ˆ KG − λ ˆ
(6.12)
2
2
1
1 2 ˆ
with λ > 0. Equation (6.12) is like ‘L ˆ = T ˆ −V ˆ ’ where V ˆ = (∇φ ˆ )
2 + m φ
2 +
2
2
λφ ˆ3 is the ‘potential’. Though simple, this Lagrangian is unfortunately not
physically sensible. The classical particle analogue potential would have the
form V (q) =
1 ωq
2 + λq
3 . If we sketch V (q) as a function of q we see that,
2
for small λ, it retains the shape of an oscillator well near q = 0, but for q
sufficiently large and negative it will ‘turn over’, tending ultimately to −∞ as
q → −∞. Classically we expect to be able to set up a successful perturbation
theory for oscillations about the equilibrium position q = 0, provided that
the amplitude of the oscillations is not so large as to carry the particle over
the ‘lip’ of the potential; in the latter case, the particle will escape to q =
−∞, invalidating a perturbative approach. In the quantum mechanical case
the same potential V (q) is more problematical, since the particle can tunnel
through the barrier separating it from the region where V → −∞. This
means that the ground state will not be stable. An analogous disease affects
the quantum field case – the supposed vacuum state will be unstable, and
indeed the energy will not be positive-definite.
Nevertheless, as the reader may already have surmised, and we shall confirm later in this chapter, the ‘φ-cubed’ interaction is precisely of the form
relevant to Yukawa’s exchange mechanism. As we have seen in the previous section, such an interaction will typically give rise to matrix elements
between one-quantum and two-quantum states, for example, exactly like the
basic Yukawa emission and absorption process. In fact, all that is necessary to make the φ ˆ3 -type interaction physical is to let it describe, not the
‘self-coupling’ of a single field, but the ‘interactive coupling’ of at least two
different fields. For example, we may have two scalar fields with quanta ‘A’
φ
2 ˆ
and ‘B’, and an interaction between them of the form λ ˆ φ B . This will allow
A